Simplifying The Expression $\frac{5(2x-3)-3(x+4)}{9}$ A Step-by-Step Solution

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In the realm of mathematics, particularly in algebra, simplifying expressions is a fundamental skill. This article delves into simplifying the algebraic expression 5(2x−3)−3(x+4)9\frac{5(2x-3)-3(x+4)}{9}, guiding you through each step to arrive at the simplest form. Mastering this process is crucial for solving more complex equations and problems in mathematics. So, let's embark on this journey of simplification and unravel the solution together!

Understanding the Basics of Algebraic Simplification

Before diving into the specific problem, it's essential to grasp the basic principles of algebraic simplification. Algebraic simplification involves reducing an algebraic expression to its simplest form while maintaining its mathematical equivalence. This often entails combining like terms, distributing constants, and applying the order of operations (PEMDAS/BODMAS). Understanding these foundational concepts is the cornerstone of successfully tackling more intricate algebraic challenges.

The Order of Operations: PEMDAS/BODMAS

At the heart of algebraic simplification lies the order of operations, commonly remembered by the acronyms PEMDAS or BODMAS. These acronyms serve as a roadmap, guiding us through the sequence in which operations must be performed to ensure accuracy. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In contrast, BODMAS represents Brackets, Orders, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Although the acronyms may differ slightly, the underlying principle remains the same: parentheses/brackets take precedence, followed by exponents/orders, then multiplication and division, and finally addition and subtraction. By adhering to this order, we can systematically simplify expressions, reducing complexity and paving the way for solutions. It's a fundamental principle that ensures consistency and accuracy in mathematical calculations, serving as a bedrock for more advanced algebraic manipulations.

Combining Like Terms: The Art of Grouping Similar Elements

In algebraic expressions, terms often come in different forms, each with its own characteristics. However, not all terms are created equal; some share similarities that allow them to be combined, thereby simplifying the expression. This process of combining like terms is a cornerstone of algebraic manipulation, enabling us to condense expressions and reveal their underlying structure. Like terms are those that contain the same variable raised to the same power. For instance, in the expression 3x+5x−2y+7y3x + 5x - 2y + 7y, the terms 3x3x and 5x5x are like terms because they both involve the variable xx raised to the power of 1. Similarly, −2y-2y and 7y7y are like terms as they both contain the variable yy raised to the power of 1. To combine like terms, we simply add or subtract their coefficients while keeping the variable and its exponent unchanged. In our example, 3x+5x3x + 5x combines to 8x8x, and −2y+7y-2y + 7y combines to 5y5y. Thus, the simplified expression becomes 8x+5y8x + 5y, a more concise and manageable form of the original. This process not only reduces the number of terms but also makes the expression easier to work with in subsequent calculations. By mastering the art of combining like terms, we gain a powerful tool for simplifying complex expressions and unlocking the secrets they hold.

Distribution: Unveiling Hidden Structures Within Expressions

Distribution is a fundamental technique in algebra that allows us to unveil the hidden structure within expressions and simplify them effectively. At its core, distribution involves multiplying a factor outside parentheses by each term inside the parentheses. This process, often referred to as the distributive property, is a cornerstone of algebraic manipulation, enabling us to expand expressions and combine like terms. Consider an expression like a(b+c)a(b + c). According to the distributive property, we multiply aa by both bb and cc, resulting in ab+acab + ac. This seemingly simple operation has profound implications, allowing us to transform complex expressions into more manageable forms. Distribution is not merely a mechanical process; it's a powerful tool for unraveling the intricacies of algebraic expressions. By applying it strategically, we can break down barriers, reveal hidden relationships, and ultimately simplify even the most daunting equations. Mastery of distribution is essential for any aspiring algebraist, as it unlocks a world of possibilities for manipulating and solving mathematical problems.

Step-by-Step Simplification of the Expression

Now, let's apply these concepts to the given expression: 5(2x−3)−3(x+4)9\frac{5(2x-3)-3(x+4)}{9}.

Step 1: Distribute the Constants

The first step involves distributing the constants outside the parentheses to the terms inside. This means multiplying 5 by both terms within the first set of parentheses (2x−3)(2x - 3) and multiplying -3 by both terms within the second set of parentheses (x+4)(x + 4). This process, known as the distributive property, is a fundamental operation in algebra that allows us to expand expressions and reveal their underlying structure. By distributing the constants, we effectively remove the parentheses, paving the way for further simplification. For instance, multiplying 5 by (2x−3)(2x - 3) yields 10x−1510x - 15, while multiplying -3 by (x+4)(x + 4) results in −3x−12-3x - 12. These expanded terms can then be combined with other terms in the expression, leading us closer to the simplest form. Mastering the art of distribution is essential for any aspiring algebraist, as it unlocks a world of possibilities for manipulating and solving mathematical problems. It's a versatile tool that empowers us to tackle complex expressions with confidence and precision.

5(2x−3)=10x−155(2x - 3) = 10x - 15

−3(x+4)=−3x−12-3(x + 4) = -3x - 12

Step 2: Rewrite the Expression

Now, substitute the results back into the expression:

10x−15−3x−129\frac{10x - 15 - 3x - 12}{9}

Step 3: Combine Like Terms

Next, we identify and combine like terms in the numerator. Like terms are terms that have the same variable raised to the same power. In this case, 10x10x and −3x-3x are like terms, and −15-15 and −12-12 are like terms. Combining like terms is a fundamental step in simplifying algebraic expressions. It allows us to condense the expression by grouping together terms that share the same variable and exponent. For example, if we have the expression 3x+2y−x+5y3x + 2y - x + 5y, we can combine the xx terms (3x3x and −x-x) to get 2x2x, and we can combine the yy terms (2y2y and 5y5y) to get 7y7y. This simplifies the expression to 2x+7y2x + 7y, which is easier to work with. In essence, combining like terms is like sorting objects into categories: we group together the apples, the oranges, and so on. This process not only makes the expression more concise but also reveals its underlying structure, making it easier to understand and manipulate. By mastering the art of combining like terms, we gain a powerful tool for simplifying complex expressions and unlocking the secrets they hold.

(10x−3x)+(−15−12)=7x−27(10x - 3x) + (-15 - 12) = 7x - 27

Step 4: Write the Simplified Expression

Substitute the combined terms back into the fraction:

7x−279\frac{7x - 27}{9}

The Simplest Form of the Expression

Therefore, the simplest form of the expression 5(2x−3)−3(x+4)9\frac{5(2x-3)-3(x+4)}{9} is 7x−279\frac{7x - 27}{9}. This corresponds to option D.

Conclusion: Mastering Algebraic Simplification

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics. By understanding the order of operations, combining like terms, and applying distribution, we can effectively reduce complex expressions to their simplest forms. This step-by-step approach not only aids in solving mathematical problems but also enhances our overall understanding of algebraic concepts. So, continue practicing and honing your skills in algebraic simplification, and you'll be well-equipped to tackle any mathematical challenge that comes your way!